Open geometry Fourier modal method: Modeling nanophotonic structures in infinite domains

We present an open geometry Fourier modal method based on a new combination of open boundary conditions and an efficient $k$-space discretization. The open boundary of the computational domain is obtained using basis functions that expand the whole space, and the integrals subsequently appearing due to the continuous nature of the radiation modes are handled using a discretization based on non-uniform sampling of the $k$-space. We apply the method to a variety of photonic structures and demonstrate that our method leads to significantly improved convergence with respect to the number of degrees of freedom, which may pave the way for more accurate and efficient modeling of open nanophotonic structures

Modeling open nanophotonic systems using the Fourier modal method: Generalization to 3D Cartesian coordinates

Recently, an open geometry Fourier modal method based on a new combination of an open boundary condition and a non-uniform $k$-space discretization was introduced for rotationally symmetric structures providing a more efficient approach for modeling nanowires and micropillar cavities [J. Opt. Soc. Am. A 33, 1298 (2016)]. Here, we generalize the approach to three-dimensional (3D) Cartesian coordinates allowing for the modeling of rectangular geometries in open space. The open boundary condition is a consequence of having an infinite computational domain described using basis functions that expand the whole space. The strength of the method lies in discretizing the Fourier integrals using a non-uniform circular "dartboard" sampling of the Fourier $k$ space. We show that our sampling technique leads to a more accurate description of the continuum of the radiation modes that leak out from the structure. We also compare our approach to conventional discretization with direct and inverse factorization rules commonly used in established Fourier modal methods. We apply our method to a variety of optical waveguide structures and demonstrate that the method leads to a significantly improved convergence enabling more accurate and efficient modeling of open 3D nanophotonic structures

Scattering from finite structures : an extended Fourier modal method

The Fourier modal method (FMM) is widely used in the diffractive optics community as an effcient tool for simulating scattering from infinitely periodic gratings. In reality the gratings are finite in size, and in applications such as lithography, it is desirable to make them as small as possible. At a certain point the assumption on infinite periodicity loses its validity. This thesis addresses the issues of extending the FMM to finite structures and consequently improving the stability and efficiency of the newly developed method. The aperiodic Fourier modal method in contrast-field formulation (AFMM-CFF) is developed by placing perfectly matched layers at the lateral sides of the computational domain and reformulating the governing equations in terms of a contrast field which does not contain the incoming field. Due to the reformulation, the homogeneous system of second-order ordinary differential equations from the original FMM becomes non-homogeneous. Its solution is derived analytically and used in the established FMM framework. The technique is first demonstrated on a simple problem of planar scattering of TE-polarized light by a single rectangular line. Later the method is generalized to arbitrary shapes of scatterers and conical incidence. The contrast-field formulation of the equations modifies the structure of the resulting linear systems and makes the direct application of available stable recursion algorithms impossible. We adapt the well-known S-matrix algorithm for use with the AFMM-CFF. To this end stable recursive relations are derived for the new type of linear systems. The stability of the algorithm is confirmed by numerical results. The effciency of the AFMM-CFF is improved by exchanging the discretization directions.Classically, spectral discretization is used in the finite periodic direction and spatial dis- cretization in the normal direction. In the light of the fact that the structures of interest have a large width-to-height ratio and that the two discretization techniques have dif- ferent computational complexities, we propose exchanging the discretization directions. This step requires a projection of the background field on the new basis introduced by the alternative discretization. For scatterers with locally repeating patterns, such as finite gratings, exchanging the discretization directions facilitates the reuse of results of previous computations, thus making the method even more efficient. As shown by numerical experiments a considerable reduction of the computational costs can be achieved

A photonic-crystal selective filter

A highly selective filter is designed, working at 1.55 μm and having a 3-dB bandwidth narrower than 0.4 nm, as is required in Dense Wavelength Division Multiplexed systems. Different solutions are proposed, involving photonic crystals made rectangular- or circular-section dielectric rods, or else of holes drilled in a dielectric bulk. The polarization and frequency selective properties are achieved by introducing a defect in the periodic structure. The device is studied by us- ing in-house codes implementing the full-wave Fourier Modal Method. Practical guidelines about advantages and limits of the investigated solutions are given

Light transmission by subwavelength square coaxial aperture arrays in metallic films

Using Fourier Modal Method, we study the enhanced transmission exhibited by arrays of square coaxial apertures in a metallic film. The calculated transmission spectrum is in good agreement with FDTD calculations. We show that the enhanced transmission can be explained considering a few guided modes of a coaxial waveguide

Design and characterisation of nanostructured gradient index lenses.

The design and characterisation of nanostructured gradient index lenses is investigated in this thesis. Nanostructured gradient index materials achieve their refractive index pro le by creating a pattern with feature sizes of =5 and smaller from two glasses with di erent refractive indices. These structures are fabricated by the stack-anddraw technology generally used for photonic crystal bres. The rigorous theoretical analysis is performed with the Fourier modal method or the nite di erence time domain algorithm. A comprehensive introduction of the Fourier modal method for one and two dimensional gratings is given. Due to the inherit periodicity of the Fourier modal method, an algorithm to calculate the transmitted eld of isolated non-periodic lamellar gratings is developed and tested experimentally with a multi layer lens grating in the microwave regime. Furthermore, the eld stitching method for the analysis of large two dimensional gratings with very small feature sizes is developed. The numerical performance is tested with a di ractive element consisting of 32 32 pixels and shown to reduce the required memory as well as the computation time by more than an order of magnitude in certain con gurations. Considerations of symmetries in the grating structure are also included in the derivation of the eld stitching method. The e ective medium theory for nanostructured gradient index materials is introduced which allows to describe nanostructured materials with the equations for standard gradient index lenses. The stack-and-draw fabrication process is described including the choice of glass types, assembly and drawing of the preforms. For the design of the required binary pattern, the simulated annealing algorithm is used in conjunction with the e ective medium theory. In order to provide experimental evidence of the simulations, two lenses were assembled from PTFE rods with a diameter of 6mm and characterised in the microwave regime at = 3 cm. It is shown that with this wavelength to feature size ratio, the nanostructured gradient index lenses can have properties nearly identical to conventional gradient index lenses. Finally, a spherical and an elliptical nanostructured microlens are characterised in the optical regime. On the elliptical microlens, phase and intensity measurements are performed and compared to simulations obtained with the Fourier modal method

On error estimation in the fourier modal method for diffractive gratings

The Fourier Modal Method (FMM, also called the Rigorous Coupled Wave Analysis, RCWA) is a numerical discretization method which is often used to calculate a scattered field from a periodic diffraction grating. For 1D periodic gratings in FMM the electromagnetic field is presented by a truncated Fourier series expansion in the direction of the grating periodicity. The grating’s material properties are assumed to be piece-wise constant (called slicing), and next per slice the scattered field is approximated by a truncated Fourier series expansion. The truncation representation of the scattered field and the piece-wise constant approximation of the grating’s material properties cause the error in FMM. This paper presents an analytical estimate/bound for the FMM error caused by slicing

Fourier modal method for inverse design of metasurface-enhanced micro-LEDs

We present a simulation capability for micro-scale light-emitting diodes (uLEDs) that achieves comparable accuracy to CPU-based finite-difference time-domain simulation but is more than 10^7 times faster. Our approach is based on the Fourier modal method (FMM) -- which, as we demonstrate, is well suited to modeling thousands of incoherent sources -- with extensions that allow rapid convergence for uLED structures that are challenging to model with standard approaches. The speed of our method makes the inverse design of uLEDs tractable, which we demonstrate by designing a metasurface-enhanced uLED that doubles the light extraction efficiency of an unoptimized device.Comment: 15 pages, 10 figure